What I believe has distinguished my own approach from a comparatively early age is a strong belief that Mathematics is very different from what we customarily imagine and that in effect the accepted paradigm represents but a reduced and very limited version of a much greater mathematical reality.

A key personal preoccupation, evident from a very early age, was the relationship as between addition and multiplication.

Before I become aware of logs (at around the age of 6), I had been experimenting with my own system of converting multiplication to addition.

Basically I operated with what would be identified as a base 2 system.

So to give a simple example, if we wished to multiply 8 by 32, the first number (in a base 2 system) could be expressed as 2

^{3 }and the second as 2

^{5 }respectively. So 8 * 32 = 2

^{3 }* 2

^{5}. Thus with respect to the RHS, we would now add the dimensional powers (indices).

Thus multiplication of the two numbers (on the LHS) would be represented as the sum of the two powers (with respect to base of 2) on the RHS.

So 8 * 32 = 2

^{3 + 5 }= 2

^{8 }= 256.

Now in principle we could express any number as a power of 2. Therefore to multiply numbers, we would express each number as 2 raised to a certain power and then add the two powers before calculating the result (i.e. 2 raised to the combined dimensional power).

I was later to discover that what I had discovered formed the very basis for logarithms.

Here, however instead of a base 2, I was initially introduced in primary school to the base 10 system.

Therefore in multiplying any two numbers, we first obtain the logs (to the base 10). In other words this expresses the power the number 10 would be raised to, in order to obtain the number in question.

Then in multiplying the two numbers we would add the logs (representing these dimensional powers).

Then to obtain our final answer we would obtain the anti-log i.e. the number that corresponds to 10 (raised to the combined sum of powers involved).

So to take our example again of 8 * 32!

In the base 10 system this would be 10

^{.9031 }* 10

^{.1.5051}.

Then adding the logs (i.e powers) we get 2.4082.

Then by obtaining the anti-log (10

^{2.4082}) we get 255.976 which approximates closely to 256.

So what fascinated me regarding this early mathematical adventure was how multiplication from one perspective (i.e. the numbers to be multiplied) represented addition (with respect to dimensional powers).

Thus I already formed an inkling of the fact that - rather than just one universal definition - numbers possessed two complementary meanings, with the first relating to a number (raised to another number) and the second relating to the corresponding power (to which the original number is raised).

Later while still attending primary school in Ireland, I began to seriously question the very manner in which multiplication is interpreted.

At the time we were working in class through simple arithmetical problems relating to the practical application of multiplication to areas and volumes.

This was before the metric era where inches, feet, yards etc provided the customary means for measuring lengths.

So if for example we had a rectangular field with length 80 yards and width 60 yards, the area of the field i.e. 4800 (i.e. just less than an acre) would be expressed in square yards.

Therefore though the starting measurements (i.e. length and width) are expressed in linear (1-dimensional) units, the corresponding area is expressed in square (2-dimensional) units.

So in multiplying the length by the width, both a quantitative and a qualitative transformation takes place with respect to the units involved.

So 80 * 60 = 4800 in quantitative terms; however we have now also switched from a linear (1-dimensional) to a square (2-dimensional) interpretation with respect to the nature of the units. In other words a qualitative transformation with respect to the (dimensional) nature of the units is likewise involved.

And here through reflection on an (apparently) simple arithmetical problem, I could see how a fundamental distinction separated the nature of addition and multiplication.

So if we seek to deal with the two lengths from an arithmetical perspective, no (qualitative) change in the nature of the units is involved.

Thus 80 + 60 (both measured in 1-dimensional units) = 140 (also measured in 1-dimensional terms). So here, the operation of addition can be interpreted in a merely quantitative manner.

However 80 * 60 (both measured in 1-dimensional units) = 4800 (now measured in 2-dimensional units).

So in contrast to addition, the operation of multiplication must be interpreted in both a quantitative and qualitative manner.

I also could see clearly at the time that a qualitative transformation would always be necessarily involved whenever multiplication is involved. With the multiplication of 2 numbers we would move to 2-dimensional units, with 3 numbers 3-dimensional units and so on!

Thus the repeated attempt, in conventional mathematical terms, to treat numbers as "abstract" quantities represents a gross form of reductionism, which thereby conceals the true nature of multiplication.

So quite simply in conventional mathematical terms, whenever numbers are multiplied, the qualitative nature of the transformation involved is reduced in a merely quantitative manner.

Thus from this perspective, 80 * 60 = 4800, i.e. 80

^{1 }* 60

^{1}

^{ }= 4800

^{1}. And strictly, this is in error. Stated in a more qualified manner, it represents but a reduced quantitative interpretation of a result that properly entails both quantitative and qualitative aspects.

I strongly realised - though of course I would have found it difficult then to properly articulate my reservations - that something truly fundamental with respect to mathematical truth was involved here and so attempted to explore the issue further.

Therefore in order to highly the key distinction involved (with respect to addition and multiplication), I concentrated on the simplest possible case.

Thus in terms of addition, 1 + 1 = 2.

Now expressed more fully with respect to linear (1-dimensional) units,

1

^{1 }+ 1

^{1}

^{ }= 2

^{1}

However, when we now apply multiplication to these two units,

1

^{1 }* 1

^{1}

^{ }= 1

^{2}.

So in the first case (with respect to addition), a quantitative transformation with respect to the units takes place (with the dimensional power or exponents of the units remaining at their default value of 1).

However by contrast (with respect to multiplication), a qualitative transformation with respect to the dimensional nature of the units takes place. (with now the base numbers remaining at their default value of 1).

What is remarkable is how the number 2 in both operations is associated with distinctive meanings!

Thus in the first case, 2 has the standard quantitative interpretation; however in the latter case 2 now relates to a distinctive qualitative interpretation!

So we have here in this simple example, the genesis for two distinctive interpretations of the number system.

The first - which I refer to as - Type 1 is of the standard quantitative nature; however the latter - Type 2 - is of a distinctive qualitative nature.

I was still too young to appreciate that these two systems implied that the number system was necessarily of a dynamic interactive nature. However fundamental progress had already certainly been made.

Now because of long training in the reduced quantitative means of interpreting mathematical symbols, the eyes of professional mathematicians will glaze over at the very mention of the qualitative which they see as having no direct relevance to their discipline.

However I wish to repeat now that qualitative notions are necessarily involved in all mathematical relationships (though consistently confused with quantitative type interpretation).

Again this is of key relevance in distinguishing the very nature of multiplication.

Stated briefly, the quantitative aspect of mathematical interpretation pertains to the treatment of numbers as separate (i.e. independent of each other). This is embodied in the cardinal notion of number.

3 for example is interpreted as a whole unit which is defined in terms of homogeneous units (that lack qualitative distinction). So 3 = 1 + 1 + 1 (with these units interpreted in an absolutely independent manner).

However the qualitative aspect of interpretation pertains to the corresponding treatment of numbers as related (i.e. interdependent with each other). This, by contrast, is embodied in the ordinal notion of number.

So if I refer now to the ordinal notion of 3 (i.e. 3rd) by definition this has no meaning in the absence of its related context with other members of a number group.

Indeed, though it is moving much further ahead, this is the basis of how all numbers can be given two distinctive meanings. So 2 for example can be given both a cardinal and ordinal meaning respectively. The cardinal then relates directly to the quantitative notion of number as separate and independent from other numbers; the ordinal then relates by contrast directly to qualitative notion of number as related and interdependent with other numbers.

Now when one reflects on experience of number, it necessarily keeps switching as between cardinal and ordinal type meaning.

The cardinal recognition of number has always an implied ordinal meaning. For example if one counts out 2 numbers (in cardinal terms) this implies recognition of a 1st and 2nd member (in an ordinal manner).

Likewise if one ranks two numbers as 1st and 2nd, then this likewise implies the cardinal recognition of 2!

However because of the dominance of the quantitative approach within Mathematics, ordinal notions are simply reduced in a cardinal manner. Thus the understanding of the primes and the natural numbers is carried out exclusively with respect to mere cardinal notions.

In this way, the key nature of the number system is thereby lost in that it properly represents the continued interaction of aspects that are both cardinal (quantitative) and ordinal (qualitative) with respect to each other.

From a cardinal perspective, each (composite) natural number represents the unique product of 2 or more primes.

So for example 6 = 2 * 3.

I have already used one type of argument to suggest that both a quantitative and qualitative transformation is involved when we multiply these two numbers.

So we can imagine 3 as independent units separated in linear time (arranged for example in a row).

So we could represent this operation by two distinct rows of 3. Then we add up all the separate units we get 6 (which represents the quantitative transformation implied by the relationship).

However the very capacity to see the two rows of 3 as common, thereby enabling the multiplication by 2, implies the corresponding recognition of interdependence (i.e. as two similar rows) which is a directly qualitative distinction.

Therefore, 2 * 3 implies recognition of both (quantitative) independence in the recognition of the separate units of each row and (qualitative) interdependent elements with respect to the recognition of the common similar nature of both rows.

So once again Conventional Mathematics inevitably reduces this interactive understanding (entailing both quantitative and qualitative aspects) in a reduced - merely - quantitative manner.

Now a deeper issue raised by this multiplication process is that the qualitative recognition of shared interdependence does not belong to linear time but in fact entails moving to an entirely new appreciation of n-dimensional time (and space). So in the simplest case involving the multiplication of two numbers, 2-dimensional time (and 2-dimensional space) is involved. Now of course because of the reduced 1-dimensional nature of Conventional Mathematics (from a qualitative perspective), we fail completely to recognise this crucially important point and thereby consistently misrepresent - in formal terms - the true dynamic nature of our experience of number.

So even at a very early age (10 or 11) I was beginning to see - literally - completely new dimensions to Mathematics that I was intent on exploring further. Thus I was already beginning to realise that what is commonly represented as Mathematics (i.e. its absolute quantitative interpretation) represents but an - admittedly very important - special case of a much more comprehensive mathematical reality.

This was brought home to me in a startling way through continual reflection on the square root of 1.

It struck me as very strange how in Conventional Mathematics the square of 1 had just one unambiguous result, whereby the corresponding square root can be given two results which are direct opposites of each other!

So the square of 1 is given as 1, whereas the two square roots of 1 are given as + 1 and – 1 respectively.

Now I reckoned that it would not be considered possible for example in the matter of mathematical proof that a proposition could equally be given a positive truth value (+ 1) and a negative truth value (– 1).

This would thereby suggest for example that the Pythagorean Theorem could be both true and false!

The very nature of accepted mathematical proof is unambiguous and designed to rule out the possibility of such a situation. Therefore by definition if a proposition is proven true, it cannot also be false (and if proven false cannot be also true). Thus in the area of proof, the positive truth value rules out the negative, and the negative truth value rules out the positive!

However here with respect to one of the simplest mathematical operations, in quantitative terms, a positive truth value is deemed to include the negative, and the negative include the positive.

For me this represented blatant inconsistency with respect to the accepted linear (1-dimensional) logic on which conventional mathematical truth is predicated.

So I already began to suspect that associated with the square of a number was a "higher" 2-dimensional logic that operated in a very different manner from what was conventionally accepted.

Furthermore I suspected that there was a direct link as between this "higher" dimensional logic (in qualitative terms) and the corresponding roots of 1 (in quantitative terms).

This was indeed heady stuff! However I lacked sufficient mental maturity (especially at a philosophical level) to make further progress at this stage.

So for the remaining primary and secondary years I maintained an uneasy balance as between conventional mathematical notions (which I already suspected were deeply flawed) and emerging personal notions (which however could not yet be properly articulated at this time).

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